## Coursera: analysis complex kind Week 3.1. Differentiability, continuity, analytic functions

### Jul 17, 2016

Key definitions in this post.

### Slide 2

Reviewed the definition of real derivative of f(x) at as limit

**Slides 3 & 4**

Graphical examples given as secant becomes tangent.

### Slide 5

“Failing” Example when graph has a “point”, that is the gradient of the curve is different each side so gradient limit from either side of x0 is not the same, therefore derivative at x0 does not exist.

### Slide 6

Complex derivative is essentially the same

… different conventions that represent derivative here ..

or, if h is a small complex number, and we write z as the limit can be written as

Since h is any complex number, can be “approached” from any “direction”.

This definition of the complex derivative as a limit with same “form” as real derivative leads to (?) the same standard results as in real-valued calculus

e.g. d/dz(c f(z)) = c f ‘(z), c is a constant

- addition rule applies
- product rule applies
- chain rule applies

as a result complicated functions, e.g. can sometimes be split into simpler components for differentiation.

## *Important* Slide 11 – non examples

Consider f(z) = Re(z)

If z = x + iy, so f(z) = Re(z) = x and let h =

(*)

So .. does this have a limit as h -> 0?

h-> 0 along **real** axis. Then

Substituting into (*) we get h / h = 1

Whereas, h-> 0 along **imaginary** axis. Then

Since Re(h) now equals zero, when we substitute this into (*) we get a quotient value of zero.

So depending on how h approaches zero (along real or imaginary axes) the difference quotient has two values, 1 and zero, so f is not differentiable at z = 0.

Further …

Consider h to be a sequence of values that spiral into towards the origin but alternating between real and imaginary values

<< sketch here would help >>

A suitable function (which we put in place of h) could be

, where n is a positive integer, creating the sequence

So that sequence of h values is spiralling towards zero

And the difference quotient (*) becomes

Which has the values

1 when n is even:

0 when n is odd

so the sequence has no limit, and since z was arbitrary, function is not differentiable anywhere on C.

## Slide 12 Another non-example f(z) =

Let f(z) =

So defining z and h as in slide 11

If h R then = 1 -> **1** as h -> 0.

If h iR then = -1 -> **-1** as h -> 0.

Thus does NOT have a limit as h -> 0, and since z is arbitrary, f (complement of z) is not differentiable **anywhere** in C.

## Slide 13 (complex) Differentiability implies continuity

## ** Slide 14 Analytic functions **

**Definition** : A function is **analytic** in an open subset U ⊂ C, if it is complex differentiable at every point in U

**Definition**: A function which is **analytic on all of C** is **entire**.

Examples:

Polynomials analytic every in C therefore entire

Rational functions p(x)/q(x) are analytic when q(x) not equal zero.

Modulus function not analytic

Re(z), Im(z) not analytic

Another example, by first principles f(z) =( modulus(z))^2 is differentiable only at the single point z=0 therefore not analytic, however it is continuous everywhere in C

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